Surface impedance inference via neural fields and sparse acoustic data obtained by a compact array

TL;DR

This work addresses the challenge of in-situ acoustic impedance characterization by reframing impedance retrieval as a physics-informed neural-field problem. A parallel multi-frequency neural field, with per-frequency SIREN-based ModMLPs and a composite loss enforcing data fidelity, Helmholtz PDE constraints, and impedance priors, enables rapid inference of complex surface impedance from sparse near-surface pressure samples. The boundary impedance emerges from the learned field via $j\zeta \partial_n p = k p$, with $\zeta = k p /(j \partial_n p)$, and is validated through numerical simulations, controlled experiments with a compact $4\times4$ MEMS array, and a virtual automotive cabin scenario. Results show robust impedance retrieval for porous absorbers and highlight the critical role of local field complexity, informing sensor layout and measurement strategies; practical guidance suggests opening enclosures to reduce nodal patterns, enabling reliable in-situ characterization for architectural and automotive acoustics across challenging environments.

Abstract

Standardized laboratory characterizations for absorbing materials rely on idealized sound field assumptions, which deviate largely from real-life conditions. Consequently, \emph{in-situ} acoustic characterization has become essential for accurate diagnosis and virtual prototyping. We propose a physics-informed neural field that reconstructs local, near-surface broadband sound fields from sparse pressure samples to directly infer complex surface impedance. A parallel, multi-frequency architecture enables a broadband impedance retrieval within runtimes on the order of seconds to minutes. To validate the method, we developed a compact microphone array with low hardware complexity. Numerical verifications and laboratory experiments demonstrate accurate impedance retrieval with a small number of sensors under realistic conditions. We further showcase the approach in a vehicle cabin to provide practical guidance on measurement locations that avoid strong interference. Here, we show that this approach offers a robust means of characterizing \emph{in-situ} boundary conditions for architectural and automotive acoustics.

Figures (4)

• Figure 1: Framework for surface impedance inference using parallel neural fields.a, Schematic concept: sparse pressure measurements near a material surface serve as input for characterizing the boundary condition. b, Spatial definition of the computational domain: boundary points ($S_b$), volumetric domain points ($S_v$), and physical sampling points ($S_s$). The sampled pressure $p(S_s, f)$ constitutes the training data target. c, Architecture of the parallel multi-frequency network and composite loss function. Independent sinusoidal representation network (SIREN) ModMLPs ($\theta_f$) predict the complex pressure $\hat{p}_f$ for each frequency $f$. The total loss $\mathcal{L}$ integrates field fidelity ($\mathcal{L}_\text{data}$: data mismatch at $S_s$; $\mathcal{L}_\text{PDE}$: Helmholtz (partial differential equation) PDE residual at $S_v$) and impedance physics ($\mathcal{L}_\text{var}$: spatial homogeneity of impedance $\zeta$; $\mathcal{L}_\text{smooth}$: frequency smoothness of the spatially-averaged reflection coefficient $\bar{R}$). Impedance $\zeta$ is defined as the normalized surface impedance $\zeta = Z/(\rho_0 c_0)$, where $\rho_0$ and $c_0$ denote the air density and sound speed. It is inferred implicitly via the boundary condition $j\zeta \frac{\partial p}{\partial n} = k p$, with $k=2\pi f/c_0$, using automatic differentiation.
• Figure 2: Numerical verification under plane-wave incidence.a, Simulation setup illustrating plane-wave incidence on a test surface under anechoic conditions, showing the two-layer sparse microphone array geometry. b, Parametric error analysis for the absorption coefficient $\alpha$. Heatmaps show the frequency-averaged mean absolute error ($\mathrm{MAE}_{\alpha}$) as a function of $(d_1,d_2)$ for the porous absorber (top) and the near-rigid surface (bottom), for 3$\times$3 and 4$\times$4 arrays. Stars mark the minimum-error configuration for each array, and the red cross indicates the practically selected configuration used for the porous absorber in subsequent analyses. c, Parametric error analysis for the normalized surface impedance $\zeta$. Heatmaps show the frequency-averaged mean absolute error ($\mathrm{MAE}_{\zeta}$) under the same geometric sweep and array configurations. d--f, Broadband inference results. d, Porous absorber using the minimum-impedance-error (starred) configurations. e, Porous absorber using the selected practical configuration (red cross). f, Near-rigid surface using the minimum-error (starred) configurations. g, Training convergence. Evolution of $\mathrm{MAE}_{\alpha}$ (top) and the normalized impedance error $\mathrm{MAE}_{\zeta}/\overline{|\zeta|}$ (bottom) versus training epochs (bottom axis) and wall-clock time (top axis), error bars denote the standard deviation across all investigated $(d_1,d_2)$ configurations. h, Sensitivity to additive noise. Heatmaps show the frequency-resolved absolute absorption error $|\Delta \alpha(f)| = |\alpha_{\mathrm{pred}}(f)-\alpha_{\mathrm{gt}}(f)|$ versus frequency and SNR for the selected configurations (porous: red cross, near-rigid: star), with columns corresponding to the 3$\times$3 and 4$\times$4 arrays.
• Figure 3: Experimental validation on finite-sized specimens across reflection-free and reverberant environments.a, Illustration of the custom-built microphone array featuring a linear sliding module and a $4\times4$ Micro-electro-mechanical systems (MEMS) microphone grid. b, Photograph of the experimental deployment showing the array positioned above a specimen with defined sampling heights ($d_1=20$ mm, $d_2=30$ mm). c, Anechoic chamber setup for characterizing angle-dependent properties. Measurements were conducted on square specimens of two edge lengths ($0.6$ m and $1.2$ m) under normal ($0^\circ$), oblique ($70^\circ$), and grazing ($85^\circ$) incidence. d, Inferred surface impedance $\zeta$ (solid/dashed lines for real/imaginary parts) and absorption coefficients $\alpha$ (insets) for the anechoic cases, compared against impedance tube data and theoretical models. e, Reverberant chamber setup testing the specimen at edge and off-center positions with varying source distances (1.5 m and 3.0 m) to modulate the direct-to-reverberant ratio. f, Inferred results in the reverberant environment: inferred impedance and random incidence absorption coefficients compared with reference values from impedance tube (measured normal incidence impedance and random incidence absorption coefficient derived via Paris' formula paris1928coefficient) and impedance model.
• Figure 4: Virtual in-situ characterization of multi-component acoustics in a full vehicle model.a, 3D visualization of the vehicle cabin sound field at 2800 Hz, highlighting the target interior components for impedance characterization. b, Comparison of the inferred broadband impedance versus ground truth references for various interiors using $3\times3$ and $4\times4$ array configurations. Shaded regions represent the standard deviation across multiple spatially distributed test positions ($N=2\text{--}4$) for each component, indicating spatial consistency. Insets display the corresponding absorption coefficients. c, Correlation between the prediction error of the pressure ($\mathrm{MAE}_p$) and the sound field complexity metric ($C$) at the door lining location (evaluated within a 20 mm height volume consisting of 52,500 points above the surface). Data are shown for $3\times3$ (circles) and $4\times4$ (squares) arrays, color-coded by frequency. d, Visualization of ground truth pressure field slices, network predictions, and corresponding error maps (real and imaginary parts) at 2400 Hz and 2700 Hz near the door lining, illustrating the two sampling layers and the impedance surface. The error maps show that sharp nodal line structures result in high-complexity regions, leading to increased inference error at the target impedance surface.